Sin, Cos, Tan , ArcSin, ArcCos, ArcTan , Exp , Ln , Sqrt , Abs , Sign , Complex , Re , Im , I , Conjugate , Arg , ! , Bin , Sum , Average , Factorize , Min , Max , IsZero , IsRational , Numer , Denom , Commutator , Taylor , InverseTaylor , ReversePoly , BigOh , Newton , D , Curl , Diverge , Integrate , Simplify , RadSimp , Rationalize , Solve , SuchThat , Eliminate , PSolve , Pi() , Random , VarList , Limit , TrigSimpCombine , LagrangeInterpolant , Fibonacci .

# Calculus and elementary functions

In this chapter, some facilities for doing calculus are described. These include functions implementing differentiation, integration, standard mathematical functions, and solving of equations.

Sin, Cos, Tan | Trigonometric functions
ArcSin, ArcCos, ArcTan | Inverse trigonometric functions
Exp | Exponential function
Ln | Natural logarithm
Sqrt | Square root
Abs | Absolute value or modulus
Sign | Sign of a number
Complex | Construct a complex number
Re | Real part of a complex number
Im | Imaginary part of a complex number
I | Imaginary unit
Conjugate | Complex conjugate
Arg | Argument of a complex number
! | Factorial
Bin | Binomial coefficients
Sum | Sum of a list of values
Average | Average of a list of values
Factorize | Product of a list of values
Min | Minimum of a number of values
Max | Maximum of a number of values
IsZero | Test whether argument is zero
IsRational | Test whether argument is a rational
Numer | Numerator of an expression
Denom | Denominator of an expression
Commutator | Commutator of two objects
Taylor | Univariate Taylor series expansion
InverseTaylor | Taylor expansion of inverse
ReversePoly | Solve h(f(x))=g(x)+O(x^n) for h
BigOh | Drop all terms of a certain order in a polynomial
Newton | Solve an equation numerically with Newton's method
D | Differentiation
Curl | Curl of a vector field
Diverge | Divergence of a vector field
Integrate | Integration
Simplify | Try to simplify an expression
RadSimp | Simplify expression with nested radicals
Rationalize | Convert floating point numbers to fractions
Solve | Solve one or more algebraic equations
SuchThat | Find a value which makes some expression zero
Eliminate | Substitute and simplify
PSolve | Solve a polynomial equation
Pi() | Numerical approximation of Pi
Random | Random number between 0 and 1
VarList | List of variables appearing in some expression
Limit | Limit of an expression
TrigSimpCombine | Combine products of trigonometric functions
LagrangeInterpolant | Polynomial interpolation
Fibonacci | Fibonacci sequence

###

Sin, Cos, Tan -- Trigonometric functions

Standard math library

##### Calling format:

    Sin(x)
    Cos(x)
    Tan(x)

##### Parameters:

`x` -- argument to the function, in radians

##### Description:

These functions represent the trigonometric functions sine, cosine, and tangent respectively. Yacas leaves them alone even if x is a number, trying to keep the result as exact as possible. The floating point approximations of these functions can be forced by using the `N` function.

Yacas knows some trigonometric identities, so it can simplify to exact results even if `N` is not used. This is the case, for instance, when the argument is a multiple of Pi/6 or Pi/4.

These functions are threaded, meaning that if the argument "x" is a list, the function is applied to all the entries in the list.

##### Examples:

    In> Sin(1)
    Out> Sin(1);
    In> N(Sin(1),20)
    Out> 0.84147098480789650665;
    In> Sin(Pi/4)
    Out> Sqrt(2)/2;

##### See also:

ArcSin , ArcCos , ArcTan , N , Pi .

###

ArcSin, ArcCos, ArcTan -- Inverse trigonometric functions

Standard math library

##### Calling format:

ArcSin(x)

ArcCos(x)

ArcTan(x)

##### Parameters:

x - argument to the function

##### Description:

These functions represent the inverse trigonometric functions. For instance, the value of "ArcSin(x)" is the number "y" such that "Sin(y)" equals "x".

Note that the number "y" is not unique. For instance, "Sin(0)" and "Sin(Pi)" both equal 0, so what should "ArcSin(0)" be? In Yacas, it is agreed that the value of "ArcSin(x)" should be in the interval [-Pi/2,Pi/2]. The same goes for "ArcTan(x)". However, "ArcCos(x)" is in the interval [0,Pi].

Usually, Yacas leaves these functions alone unless it is forced to do a numerical evaluation by the `N` function. If the argument is -1, 0, or 1 however, Yacas will simplify the expression. If the argument is complex, the expression will be rewritten as a `Ln` function.

These functions are threaded, meaning that if the argument "x" is a list, the function is applied to all the entries in the list.

##### Examples:

    In> ArcSin(1)
    Out> Pi/2;

    In> ArcSin(1/3)
    Out> ArcSin(1/3);
    In> Sin(ArcSin(1/3))
    Out> 1/3;

    In> N(ArcSin(0.75))
    Out> 0.848062;
    In> N(Sin(%))
    Out> 0.7499999477;

##### See also:

Sin , Cos , Tan , N , Pi , Ln .

###

Exp -- Exponential function

Standard math library

##### Calling format:

    Exp(x)

##### Parameters:

x - argument to the function

##### Description:

This function calculates e raised to the power "x", where e is the mathematic constant 2.71828... One can use `Exp(1)` to represent e.

This function is threaded, meaning that if the argument "x" is a list, the function is applied to all the entries in the list.

##### Examples:

    In> Exp(0)
    Out> 1;
    In> Exp(I*Pi)
    Out> -1;
    In> N(Exp(1))
    Out> 2.7182818284;

##### See also:

Ln , Sin , Cos , Tan , N .

###

Ln -- Natural logarithm

Standard math library

##### Calling format:

    Ln(x)

##### Parameters:

x - argument to the function

##### Description:

This function calculates the natural logarithm of "x". This is the inverse function of the exponential function, `Exp`, ie. "Ln(x) = y" implies that "Exp(y) = x". For complex arguments, the imaginary part of the logarithm is in the interval (- Pi, Pi]. This is compatible with the branch cut of `Arg`.

This function is threaded, meaning that if the argument "x" is a list, the function is applied to all the entries in the list.

##### Examples:

    In> Ln(1)
    Out> 0;
    In> Ln(Exp(x))
    Out> x;
    In> D(x) Ln(x)
    Out> 1/x;

##### See also:

Exp , Arg .

###

Sqrt -- Square root

Standard math library

##### Calling format:

    Sqrt(x)

##### Parameters:

x - argument to the function

##### Description:

This function calculates the square root of "x". If the result is not rational, the call is returned unevaluated unless a numerical approximation is forced with the `N` function. This function can also handle negative and complex arguments.

This function is threaded, meaning that if the argument "x" is a list, the function is applied to all the entries in the list.

##### Examples:

    In> Sqrt(16)
    Out> 4;
    In> Sqrt(15)
    Out> Sqrt(15);
    In> N(Sqrt(15))
    Out> 3.8729833462;
    In> Sqrt(4/9)
    Out> 2/3;
    In> Sqrt(-1)
    Out> Complex(0,1);

##### See also:

Exp , ^ , N .

###

Abs -- Absolute value or modulus

Standard math library

##### Calling format:

    Abs(x)

##### Parameters:

x - argument to the function

##### Description:

This function returns the absolute value (also called the modulus) of "x". If "x" is positive, the absolute value is "x" itself; if "x" is negative, the absolute value is "-x". For complex "x", the modulus is the "r" in the polar decomposition x=r*Exp(I*phi).

This function is connected to the `Sign` function by the identity "Abs(x) * Sign(x) = x" for real "x".

This function is threaded, meaning that if the argument "x" is a list, the function is applied to all the entries in the list.

##### Examples:

    In> Abs(2);
    Out> 2;
    In> Abs(-1/2);
    Out> -1/2;
    In> Abs(3+4*I);
    Out> 5;

##### See also:

Sign , Arg .

###

Sign -- Sign of a number

Standard math library

##### Calling format:

Sign(x)

##### Parameters:

x - argument to the function

##### Description:

This function returns the sign of the real number "x". It is "1" for positive numbers and "-1" for negative numbers. Somewhat arbitrarily, `Sign(0)` is defined to be 1.

This function is connected to the `Abs` function by the identity "Abs(x) * Sign(x) = x" for real "x".

This function is threaded, meaning that if the argument "x" is a list, the function is applied to all the entries in the list.

##### Examples:

    In> Sign(2)
    Out> 1;
    In> Sign(-3)
    Out> -1;
    In> Sign(0)
    Out> 1;
    In> Sign(-3) * Abs(-3)
    Out> -3;

##### See also:

Arg , Abs .

###

Complex -- Construct a complex number

Standard math library

##### Calling format:

    Complex(r, c)

##### Parameters:

r - real part

c - imaginary part

##### Description:

This function represents the complex number "r + I*c", where "I" is the imaginary unit. It is the standard representation used in Yacas to represent complex numbers. Both "r" and "c" are supposed to be real.

Note that, at the moment, many functions in Yacas assume that all numbers are real unless it is obvious that it is a complex number. Hence `Im(Sqrt(x))` evaluates to `0` which is only true for nonnegative "x".

##### Examples:

    In> I
    Out> Complex(0,1);
    In> 3+4*I
    Out> Complex(3,4);
    In> Complex(-2,0)
    Out> -2;

##### See also:

Re , Im , I , Abs , Arg .

###

Re -- Real part of a complex number

Standard math library

##### Calling format:

Re(x)

##### Parameters:

x - argument to the function

##### Description:

This function returns the real part of the complex number "x".

##### Examples:

    In> Re(5)
    Out> 5;
    In> Re(I)
    Out> 0;
    In> Re(Complex(3,4))
    Out> 3;

##### See also:

Complex , Im .

###

Im -- Imaginary part of a complex number

Standard math library

##### Calling format:

    Im(x)

##### Parameters:

x - argument to the function

##### Description:

This function returns the imaginary part of the complex number "x".

##### Examples:

    In> Im(5)
    Out> 0;
    In> Im(I)
    Out> 1;
    In> Im(Complex(3,4))
    Out> 4;

##### See also:

Complex , Re .

###

I -- Imaginary unit

Standard math library

##### Calling format:

    I

##### Description:

This symbol represents the imaginary unit, which equals the square root of -1. It evaluates to `Complex(0,1)`.

##### Examples:

    In> I
    Out> Complex(0,1);
    In> I = Sqrt(-1)
    Out> True;

##### See also:

Complex .

###

Conjugate -- Complex conjugate

Standard math library

##### Calling format:

    Conjugate(x)

##### Parameters:

x - argument to the function

##### Description:

This function returns the complex conjugate of "x". The complex conjugate of a+I*b is a-I*b. This function assumes that all unbound variables are real.

##### Examples:

    In> Conjugate(2)
    Out> 2;
    In> Conjugate(Complex(a,b))
    Out> Complex(a,-b);

##### See also:

Complex , Re , Im .

###

Arg -- Argument of a complex number

Standard math library

##### Calling format:

    Arg(x)

##### Parameters:

x - argument to the function

##### Description:

This function returns the argument of "x". The argument is the angle with the positive real axis in the Argand diagram, or the angle "phi" in the polar representation r*Exp(I*phi) of "x". The result is in the range (-Pi, Pi], that is, excluding -Pi but including Pi. The argument of 0 is `Undefined`.

##### Examples:

    In> Arg(2)
    Out> 0;
    In> Arg(-1)
    Out> Pi;
    In> Arg(1+I)
    Out> Pi/4;

##### See also:

Abs , Sign .

###

! -- Factorial

Standard math library

##### Calling format:

    n!

##### Parameters:

n - argument to the function

##### Description:

This function calculates the factorial of integer or half-integer numbers. For nonnegative integers, n! equals n*(n-1)*(n-2)*...*1. The factorial of half-integers, defined via the gamma function, is also evaluated.

This function is threaded, meaning that if the argument "x" is a list, the function is applied to all the entries in the list.

For reasons of Yacas syntax, the factorial sign `!` cannot precede other non-letter symbols such as `+` or `*`. Therefore, you should enter a space after `!` in expressions such as `x! +1`.

##### Examples:

    In> 5!
    Out> 120;
    In> 1 * 2 * 3 * 4 * 5
    Out> 120;
    In> (1/2)!
    Out> Sqrt(Pi)/2;

##### See also:

Bin , Factorize .

###

Bin -- Binomial coefficients

Standard math library

##### Calling format:

Bin(n, m)

##### Parameters:

n, m - integers

##### Description:

This function calculates the binomial coefficient "n" above "m", which equals

n! /(n! *(n-m)!)

This is equal to the number of ways to choose "m" objects out of a total of "n" objects if order is not taken into account. The binomial coefficient is defined to be zero if "m" is negative or greater than "n".

##### Examples:

    In> Bin(10, 4)
    Out> 210;
    In> 10! / (4! * 6!)
    Out> 210;

##### See also:

! .

###

Sum -- Sum of a list of values

Standard math library

##### Calling format:

Sum(list)

Sum(var, from, to, body)

##### Parameters:

list - list of values to sum

var - variable to iterate over

from - integer value to iterate from

to - integer value to iterate up to

body - expression to evaluate for each iteration

##### Description:

The first form of the `Sum` command simply adds all the entries in "list" and returns their sum.

If the second calling sequence is used, the expression "body" is evaluated while the variable "var" ranges over all integers from "from" up to "to", and the sum of all the results is returned. Obviously, "to" should be greater than or equal to "from".

##### Examples:

    In> Sum({1,4,9});
    Out> 14;
    In> Sum(i, 1, 3, i^2);
    Out> 14;

##### See also:

Average , Factorize , Apply .

###

Average -- Average of a list of values

Standard math library

##### Calling format:

Average(list)

##### Parameters:

list - list of values to average

##### Description:

This command calculates the (arithmetical) average of all the entries in "list", which is the sum of all entries divided by the number of entries.

##### Examples:

    In> Average({1,2,3,4,5});
    Out> 3;
    In> Average({2,6,7});
    Out> 5;

##### See also:

Sum .

###

Factorize -- Product of a list of values

Standard math library

##### Calling format:

Factorize(list)

Factorize(var, from, to, body)

##### Parameters:

list - list of values to multiply

var - variable to iterate over

from - integer value to iterate from

to - integer value to iterate up to

body - expression to evaluate for each iteration

##### Description:

The first form of the `Factorize` command simply multiplies all the entries in "list" and returns their product.

If the second calling sequence is used, the expression "body" is evaluated while the variable "var" ranges over all integers from "from" up to "to", and the product of all the results is returned. Obviously, "to" should be greater than or equal to "from".

##### Examples:

    In> Factorize({1,2,3,4});
    Out> 24;
    In> Factorize(i, 1, 4, i);
    Out> 24;

##### See also:

Sum , Apply .

###

Min -- Minimum of a number of values

Standard math library

##### Calling format:

Min(x,y)

Min(list)

##### Parameters:

x, y - pair of values to determine the minimum of

list - list of values from which the minimum is sought

##### Description:

This function returns the minimum value of its argument(s). If the first calling sequence is used, the smaller of "x" and "y" is returned. If one uses the second form, the smallest of the entries in "list" is returned. In both cases, this function can only be used with numerical values and not with symbolic arguments.

##### Examples:

    In> Min(2,3);
    Out> 2;
    In> Min({5,8,4});
    Out> 4;

##### See also:

Max , Sum , Average .

###

Max -- Maximum of a number of values

Standard math library

##### Calling format:

Max(x,y)

Max(list)

##### Parameters:

x, y - pair of values to determine the maximum of

list - list of values from which the maximum is sought

##### Description:

This function returns the maximum value of its argument(s). If the first calling sequence is used, the larger of "x" and "y" is returned. If one uses the second form, the largest of the entries in "list" is returned. In both cases, this function can only be used with numerical values and not with symbolic arguments.

##### Examples:

    In> Max(2,3);
    Out> 3;
    In> Max({5,8,4});
    Out> 8;

##### See also:

Min , Sum , Average .

###

IsZero -- Test whether argument is zero

Standard math library

##### Calling format:

IsZero(n)

##### Parameters:

n - number to test

##### Description:

`IsZero(n)` evaluates to `True` if "n" is zero. In case "n" is not a number, the function returns `False`.

##### Examples:

    In> IsZero(3.25)
    Out> False;
    In> IsZero(0)
    Out> True;
    In> IsZero(x)
    Out> False;

##### See also:

IsNumber , IsNotZero .

###

IsRational -- Test whether argument is a rational

Standard math library

##### Calling format:

IsRational(expr)

##### Parameters:

expr - expression to test

##### Description:

This commands tests whether the expression "expr" is a rational number. This is the case if the top-level operator of "expr" is `/`.

##### Examples:

    In> IsRational(5)
    Out> False;
    In> IsRational(2/7)
    Out> True;
    In> IsRational(a/b)
    Out> True;
    In> IsRational(x + 1/x)
    Out> False;

##### See also:

Numer , Denom .

###

Numer -- Numerator of an expression

Standard math library

##### Calling format:

Numer(expr)

##### Parameters:

expr - expression to determine numerator of

##### Description:

This function determines the numerator of the rational expression "expr" and returns it. As a special case, if its argument is numeric but not rational, it returns this number. If "expr" is neither rational nor numeric, the function returns unevaluated.

##### Examples:

    In> Numer(2/7)
    Out> 2;
    In> Numer(a / x^2)
    Out> a;
    In> Numer(5)
    Out> 5;

##### See also:

Denom , IsRational , IsNumber .

###

Denom -- Denominator of an expression

Standard math library

##### Calling format:

Denom(expr)

##### Parameters:

expr - expression to determine denominator of

##### Description:

This function determines the denominator of the rational expression "expr" and returns it. As a special case, if its argument is numeric but not rational, it returns `1`. If "expr" is neither rational nor numeric, the function returns unevaluated.

##### Examples:

    In> Denom(2/7)
    Out> 7;
    In> Denom(a / x^2)
    Out> x^2;
    In> Denom(5)
    Out> 1;

##### See also:

Numer , IsRational , IsNumber .

###

Commutator -- Commutator of two objects

Standard math library

##### Calling format:

Commutator(a, b)

##### Parameters:

a, b - two objects whose commutator should be computed

##### Description:

This command computes the commutator of 'a" and "b", ie. the expression "a b - b a". For numbers and other objects for which multiplication is commutative, the commutator is zero. But this is not necessarily the case for matrices.

##### Examples:

    In> Commutator(2,3)
    Out> 0;
    In> PrettyPrinter("PrettyForm");

    True

    Out>
    In> A := { {0,x}, {0,0} }

    /              \
    | ( 0 ) ( x )  |
    |              |
    | ( 0 ) ( 0 )  |
    \              /

    Out>
    In> B := { {y,0}, {0,z} }

    /              \
    | ( y ) ( 0 )  |
    |              |
    | ( 0 ) ( z )  |
    \              /

    Out>
    In> Commutator(A,B)

    /                          \
    | ( 0 ) ( x * z - y * x )  |
    |                          |
    | ( 0 ) ( 0 )              |
    \                          /

    Out>

###

Taylor -- Univariate Taylor series expansion

Standard math library

##### Calling format:

Taylor(var, at, order) expr

##### Parameters:

var - variable

at - point to get Taylor series around

order - order of approximation

expr - expression to get Taylor series for

##### Description:

This function returns the Taylor series expansion of the expression "expr" with respect to the variable "var" around "at" up to order "order". This is a polynomial which agrees with "expr" at the point "var = at", and furthermore the first "order" derivatives of the polynomial at this point agree with "expr". Taylor expansions around removable singularities are correctly handled by taking the limit as "var" approaches "at".

##### Examples:

    In> PrettyForm(Taylor(x,0,9) Sin(x))

         3    5      7       9
        x    x      x       x
    x - -- + --- - ---- + ------
        6    120   5040   362880

    Out> True;

##### See also:

D , InverseTaylor , ReversePoly , BigOh .

###

InverseTaylor -- Taylor expansion of inverse

Standard math library

##### Calling format:

InverseTaylor(var, at, order) expr

##### Parameters:

var - variable

at - point to get inverse Taylor series around

order - order of approximation

expr - expression to get inverse Taylor series for

##### Description:

This function builds the Taylor series expansion of the inverse of the expression "expr" with respect to the variable "var" around "at" up to order "order". It uses the function `ReversePoly` to perform the task.

##### Examples:

    In> PrettyPrinter("PrettyForm")

    True

    Out>
    In> exp1 := Taylor(x,0,7) Sin(x)

         3    5      7
        x    x      x
    x - -- + --- - ----
        6    120   5040

    Out>
    In> exp2 := InverseTaylor(x,0,7) ArcSin(x)

     5      7     3
    x      x     x
    --- - ---- - -- + x
    120   5040   6

    Out>
    In> Simplify(exp1-exp2)

    0

    Out>

##### See also:

ReversePoly , Taylor , BigOh .

###

ReversePoly -- Solve h(f(x))=g(x)+O(x^n) for h

Standard math library

##### Calling format:

ReversePoly(f, g, var, newvar, degree)

##### Parameters:

f, g - expressions in "var"

var - a variable

newvar - a new variable to express the result in

degree - the degree of the required solution

##### Description:

This function returns a polynomial in "newvar", say "h(newvar)", with the property that "h(f(var))" equals "g(var)" up to order "degree". The degree of the result will be at most "degree-1". The only requirement is that the first derivative of "f" should not be zero.

This function is used to determine the Taylor series expansion of the inverse of a function "f": if we take "g(var)=var", then "h(f(var))=var" (up to order "degree"), so "h" will be the inverse of "f".

##### Examples:

    In> f(x):=Eval(Expand((1+x)^4))
    Out> True;
    In> g(x) := x^2
    Out> True;
    In> h(y):=Eval(ReversePoly(f(x),g(x),x,y,8))
    Out> True;
    In> BigOh(h(f(x)),x,8)
    Out> x^2;
    In> h(x)
    Out> (-2695*(x-1)^7)/131072+(791*(x-1)^6)
    /32768 +(-119*(x-1)^5)/4096+(37*(x-1)^4)
    /1024+(-3*(x-1)^3)/64+(x-1)^2/16;

##### See also:

InverseTaylor , Taylor , BigOh .

###

BigOh -- Drop all terms of a certain order in a polynomial

Standard math library

##### Calling format:

BigOh(poly, var, degree)

##### Parameters:

poly - a univariate polynomial

var - a free variable

degree - positive integer

##### Description:

This function drops all terms of order "degree" or higher in "poly", which is a polynomial in the variable "var".

##### Examples:

    In> BigOh(1+x+x^2+x^3,x,2)
    Out> x+1;

##### See also:

Taylor , InverseTaylor .

###

Newton -- Solve an equation numerically with Newton's method

Standard math library

##### Calling format:

Newton(expr, var, initial, accuracy)

##### Parameters:

expr - an expression to find a zero for

var - free variable to adjust to find a zero

initial - initial value for "var" to use in the search

accuracy - minimum required accuracy of the result

##### Description:

This function tries to numerically find a zero of the expression "expr", which should depend only on the variable "var". It uses the value "initial" as an initial guess.

The function will iterate using Newton's method until it estimates that it has come within a distance "accuracy" of the correct solution, and then it will return its best guess. In particular, it may loop forever if the algorithm does not converge.

##### Examples:

    In> Newton(Sin(x),x,3,0.0001)
    Out> 3.1415926535;

##### See also:

Solve .

###

D -- Differentiation

Standard math library

##### Calling format:

D(var) expr

D(list) expr

D(var,n) expr

##### Parameters:

var - variable

list - a list of variables

expr - expression to take derivative of

n - order of derivative

##### Description:

This function calculates the derivative of the expression "expr" with respect to the variable "var" and returns it. If the third calling sequence is used, the "n"-th derivative is determined. Yacas knows how the differentiate standard functions like `Ln` and `Sin`.

The `D` operator is threaded in both "var" and "expr". This means that if either of them is a list, the function is applied to each entry in the list. The results are collected in another list which is returned. If both "var" and "expr" are a list, their lengths should be equal. In this case, the first entry in the list "expr" is differentiated with respect to the first entry in the list "var", the second entry in "expr" is differentiated with respect to the second entry in "var", and so on.

##### Examples:

    In> D(x)Sin(x*y)
    Out> y*Cos(x*y);
    In> D({x,y,z})Sin(x*y)
    Out> {y*Cos(x*y),x*Cos(x*y),0};
    In> D(x,2)Sin(x*y)
    Out> -Sin(x*y)*y^2;
    In> D(x){Sin(x),Cos(x)}
    Out> {Cos(x),-Sin(x)};

##### See also:

Integrate , Taylor , Diverge , Curl .

###

Curl -- Curl of a vector field

Standard math library

##### Calling format:

Curl(vector, basis)

##### Parameters:

vector - vector field to take the curl of

basis - list of variables forming the basis

##### Description:

This function takes the curl of the vector field "vector" with respect to the variables "basis". The curl is defined in the usual way,

    Curl(f,x) = {
        D(x[2]) f[3] - D(x[3]) f[2],
        D(x[3]) f[1] - D(x[1]) f[3],
        D(x[1]) f[2] - D(x[2]) f[1]
    }

Both "vector" and "basis" should be lists of length 3.

##### Example:

    In> Curl({x*y,x*y,x*y},{x,y,z})
    Out> {x,-y,y-x};

##### See also:

D , Diverge .

###

Diverge -- Divergence of a vector field

Standard math library

##### Calling format:

Diverge(vector, basis)

##### Parameters:

vector - vector field to calculate the divergence of

basis - list of variables forming the basis

##### Description:

This function calculates the divergence of the vector field "vector" with respect to the variables "basis". The divergence is defined as

    Diverge(f,x) = D(x[1]) f[1] + ...
        + D(x[n]) f[n],

where `n` is the length of the lists "vector" and "basis". These lists should have equal length.

##### Example:

    In> Diverge({x*y,x*y,x*y},{x,y,z})
    Out> y+x;

##### See also:

D , Curl .

###

Integrate -- Integration

Standard math library

##### Calling format:

Integrate(var, from, to) expr

Integrate(var) expr

##### Parameters:

var - variable to integrate over

from - begin of interval to integrate over

to - end of interval to integrate over

expression - expression to integrate

##### Description:

This function integrates the expression "expr" with respect to the variable "var". The first calling sequence is used to perform definite integration: the integration is carried out from "var=form" to "var=to". The second form signifies indefinite integration. In this case, the function `UniqueConstant` is called to get a variable of the form `C`n (where "n" is an integer) which represent the integration constant.

Some simple integration rules have currently been implemented. Polynomials, quotients of polynomials, the transcendental functions `Sin`, `Cos`, `Exp`, and `Ln`, and products of these functions with polynomials can all be integrated.

##### Examples:

    In> Integrate(x,a,b) Cos(x)
    Out> Sin(b)-Sin(a);
    In> Integrate(x) Cos(x)
    Out> Sin(x)+C9;

##### See also:

D , UniqueConstant .

###

Simplify -- Try to simplify an expression

Standard math library

##### Calling format:

Simplify(expr)

##### Parameters:

expression - expression to simplify

##### Description:

This function tries to simplify the expression "expr" as much as possible. It does this by grouping powers within terms, and then grouping like terms.

##### Examples:

    In> a*b*a^2/b-a^3
    Out> (b*a^3)/b-a^3;
    In> Simplify(a*b*a^2/b-a^3)
    Out> 0;

##### See also:

TrigSimpCombine , RadSimp .

###

RadSimp -- Simplify expression with nested radicals

Standard math library

##### Calling format:

RadSimp(expr)

##### Parameters:

expr - an expression containing nested radicals

##### Description:

This function tries to write the expression "expr" as a sum of roots of integers: Sqrt(e1)+Sqrt(e2)+..., where e1, e2 and so on are natural numbers. The expression "expr" may not contain free variables.

It does this by trying all possible combinations for e1, e2, ... Every possibility is numerically evaluated using `N` and compared with the numerical evaluation of "expr". If the approximations are equal (up to a certain margin), this possibility is returned. Otherwise, the expression is returned unevaluated.

Note that due to the use of numerical approximations, there is a small chance that the expression returned by `RadSimp` is close but not equal to "expr". The last example underneath illustrates this problem. Furthermore, if the numerical value of "expr" is large, the number of possibilities becomes exorbitantly big so the evaluation may take very long.

##### Examples:

    In> RadSimp(Sqrt(9+4*Sqrt(2)))
    Out> Sqrt(8)+1;
    In> RadSimp(Sqrt(5+2*Sqrt(6)) \
      +Sqrt(5-2*Sqrt(6)))
    Out> Sqrt(12);
    In> RadSimp(Sqrt(14+3*Sqrt(3+2
    *Sqrt(5-12*Sqrt(3-2*Sqrt(2))))))
    Out> Sqrt(2)+3;

But this command may yield incorrect results:

    In> RadSimp(Sqrt(1+10^(-6)))
    Out> 1;

##### See also:

Simplify , N .

###

Rationalize -- Convert floating point numbers to fractions

Standard math library

##### Calling format:

Rationalize(expr)

##### Parameters:

expr - an expression containing real numbers

##### Description:

This command converts every real number in the expression "expr" into a rational number. This is useful when a calculation needs to be done on floating point numbers and the algorithm is unstable. Converting the floating point numbers to rational numbers will force calculations to be done with infinite precision (by using rational numbers as representations).

It does this by finding the smallest integer n such that multiplying the number with 10^n is an integer. Then it divides by 10^n again, depending on the internal gcd calculation to reduce the resulting division of integers.

##### Examples:

    In> {1.2, 3.123, 4.5}
    Out> {1.2,3.123,4.5};
    In> Rationalize(%)
    Out> {6/5,3123/1000,9/2};

##### See also:

IsRational .

###

Solve -- Solve one or more algebraic equations

Standard math library

##### Calling format:

Solve(eq, var)

Solve(eqlist, varlist)

##### Parameters:

eq - single identity equation

var - single variable

eqlist - list of identity equations

varlist - list of variables

##### Description:

This command tries to solve one or more equations. Use the first form to solve a single equation and the second one for systems of equations.

The first calling sequence solves the equation "eq" for the variable "var". Use the `==` operator to form the equation. The value of "var" which satisfies the equation, is returned. Note that only one solution is found and returned.

To solve a system of equations, the second form should be used. It solves the system of equations contained in the list "eqlist" for the variables appearing in the list "varlist". A list of results is returned, and each result is a list containing the values of the variables in "varlist". Again, at most a single solution is returned.

The task of solving a single equation is simply delegated to `SuchThat`. Multiple equations are solved recursively: firstly, an equation is sought in which one of the variables occurs exactly once; then this equation is solved with `SuchThat`; and finally the solution is substituted in the other equations by `Eliminate` decreasing the number of equations by one. This suffices for all linear equations and a large group of simple nonlinear equations.

##### Examples:

    In> Solve(a+x*y==z,x)
    Out> (z-a)/y;
    In> Solve({a*x+y==0,x+z==0},{x,y})
    Out> {{-z,z*a}};

This means that "x = (z-a)/y" is a solution of the first equation and that "x = -z", "y = z*a" is a solution of the systems of equations in the second command.

An example which `Solve` cannot solve:

    In> Solve({x^2-x == y^2-y,x^2-x == y^3+y},{x,y});
    Out> {};

##### See also:

SuchThat , Eliminate , PSolve , == .

###

SuchThat -- Find a value which makes some expression zero

Standard math library

##### Calling format:

    SuchThat(expr, var)

##### Parameters:

expr - expression to make zero

var - variable (or subexpression) to solve for

##### Description:

This functions tries to find a value of the variable "var" which makes the expression "expr" zero. It is also possible to pass a subexpression as "var", in which case `SuchThat` will try to solve for that subexpression.

Basically, only expressions in which "var" occurs only once are handled; in fact, `SuchThat` may even give wrong results if the variables occurs more than once. This is a consequence of the implementation, which repeatedly applies the inverse of the top function until the variable "var" is reached.

##### Examples:

    In> SuchThat(a+b*x, x)
    Out> (-a)/b;
    In> SuchThat(Cos(a)+Cos(b)^2, Cos(b))
    Out> Cos(a)^(1/2);
    In> Expand(a*x+b*x+c, x)
    Out> (a+b)*x+c;
    In> SuchThat(%, x)
    Out> (-c)/(a+b);

##### See also:

Solve , Subst , Simplify .

###

Eliminate -- Substitute and simplify

Standard math library

##### Calling format:

    Eliminate(var, value, expr)

##### Parameters:

var - variable (or subexpression) to substitute

value - new value of "var"

expr - expression in which the substitution should take place

##### Description:

This function uses `Subst` to replace all instances of the variable (or subexpression) "var" in the expression "expr" with "value", calls `Simplify` to simplify the resulting expression, and returns the result.

##### Examples:

    In> Subst(Cos(b), c) (Sin(a)+Cos(b)^2/c)
    Out> Sin(a)+c^2/c;
    In> Eliminate(Cos(b), c, Sin(a)+Cos(b)^2/c)
    Out> Sin(a)+c;

##### See also:

SuchThat , Subst , Simplify .

###

PSolve -- Solve a polynomial equation

Standard math library

##### Calling format:

    PSolve(poly, var)

##### Parameters:

poly - a polynomial in "var"

var - a variable

##### Description:

This commands returns a list containing the roots of "poly", considered as a polynomial in the variable "var". If there is only one root, it is not returned as a one-entry list but just by itself. A double root occurs twice in the result, and similarly for roots of higher multiplicity. All polynomials of degree up to 4 are handled.

##### Examples:

    In> PSolve(b*x+a,x)
    Out> -a/b;
    In> PSolve(c*x^2+b*x+a,x)
    Out> {(Sqrt(b^2-4*c*a)-b)/(2*c),(-(b+
    Sqrt(b^2-4*c*a)))/(2*c)};

##### See also:

Solve , Factor .

###

Pi() -- Numerical approximation of Pi

Internal function

##### Calling format:

    Pi()

##### Description:

This commands returns the value of the mathematical constant pi at the current precision, as set by `Precision`. Usually this function should not be called directly. The constant `Pi` can (and should) be used to represent the exact value of pi, as it is recognized by the simplification rules. When the function `N` is invoked on an expression, `Pi` will be replaced with the value returned by `Pi()`.

##### Examples:

    In> Pi()
    Out> 3.14159265358979323846;
    In> Sin(3*Pi/2)
    Out> -1;
    In> Sin(3*Pi()/2)
    Out> Sin(4.7123889804);
    In> Precision(35)
    Out> True;
    In> Pi()
    Out> 3.14159265358979323846264338327950288;

##### See also:

N , Pi , Precision .

###

Random -- Random number between 0 and 1

Standard math library

##### Calling format:

    Random()

##### Description:

This function returns a random number, uniformly distributed in the interval between 0 and 1. The same sequence of random numbers is generated in every Yacas session.

##### See also:

RandomInteger , RandomPoly .

###

VarList -- List of variables appearing in some expression

Standard math library

##### Calling format:

    VarList(expr)

##### Parameters:

expr - an expression

##### Description:

This command returns a list of all the variables that appear in the expression "expr".

##### Examples:

    In> VarList(Sin(x))
    Out> {x};
    In> VarList(x+a*y)
    Out> {x,a,y};

##### See also:

IsFreeOf , IsVariable .

###

Limit -- Limit of an expression

Standard math library

##### Calling format:

    Limit(var, val) expr
    Limit(var, val, dir) expr

##### Parameters:

var - a variable

val - a number

dir - a direction (`Left` or `Right`)

expr - an expression

##### Description:

This command tries to determine the value that the expression "expr" converges to when the variable "var" approaches "val". One may use `Infinity` or `-Infinity` for "val". The result of `Limit` may be one of the symbols `Undefined` (meaning that the limit does not exist), `Infinity`, or `-Infinity`.

The second calling sequence is used for unidirectional limits. If one gives "dir" the value `Left`, the limit is taken as "var" approaches "val" from the positive infinity; and `Right` will take the limit from the negative infinity.

##### Examples:

    In> Limit(x,0) Sin(x)/x
    Out> 1;
    In> Limit(x,0) (Sin(x)-Tan(x))/(x^3)
    Out> -1/2;
    In> Limit(x,0) 1/x
    Out> Undefined;
    In> Limit(x,0,Left) 1/x
    Out> -Infinity;
    In> Limit(x,0,Right) 1/x
    Out> Infinity;

###

TrigSimpCombine -- Combine products of trigonometric functions

Standard math library

##### Calling format:

TrigSimpCombine(expr)

##### Parameters:

expr - expression to simplify

##### Description:

This function applies the product rules of trigonometry, like Cos(u)*Sin(v)=1/2*(Sin(v-u)+Sin(v+u)). As a result, all products of the trigonometric functions `Cos` and `Sin` disappear. The function also tries to simplify the resulting expression as much as possible by combining all like terms.

This function is used in for instance `Integrate`, to bring down the expression into a simpler form that hopefully can be integrated easily.

##### Examples:

    In> PrettyPrinter("PrettyForm");

    True

    Out>
    In> TrigSimpCombine(Cos(a)^2+Sin(a)^2)

    1

    Out>
    In> TrigSimpCombine(Cos(a)^2-Sin(a)^2)

    Cos( -2 * a )

    Out>
    In> TrigSimpCombine(Cos(a)^2*Sin(b))

    Sin( b )   Sin( -2 * a + b )
    -------- + -----------------
       2               4

        Sin( -2 * a - b )
      - -----------------
                4

    Out>

##### See also:

Simplify , Integrate , Expand , Sin , Cos , Tan .

###

LagrangeInterpolant -- Polynomial interpolation

Standard math library

##### Calling format:

LagrangeInterpolant(xlist, ylist, var)

##### Parameters:

xlist - list of argument values

ylist - list of function values

var - free variable for resulting polynomial

##### Description:

This function returns a polynomial in the variable "var" which interpolates the points "(xlist, ylist)". Specifically, the value of the resulting polynomial at "xlist[1]" is "ylist[1]", the value at "xlist[2]" is "ylist[2]", etc. The degree of the polynomial is not greater than the length of "xlist".

The lists "xlist" and "ylist" should be of equal length. Furthermore, the entries of "xlist" should be all distinct to ensure that there is one and only one solution.

This routine uses the Lagrange interpolant formula to build up the polynomial.

##### Examples:

    In> f := LagrangeInterpolant({0,1,2}, \
      {0,1,1}, x);
    Out> (x*(x-1))/2-x*(x-2);
    In> Eval(Subst(x,0) f);
    Out> 0;
    In> Eval(Subst(x,1) f);
    Out> 1;
    In> Eval(Subst(x,2) f);
    Out> 1;

    In> PrettyPrinter("PrettyForm");

    True

    Out>
    In> LagrangeInterpolant({x1,x2,x3}, {y1,y2,y3}, x)

    y1 * ( x - x2 ) * ( x - x3 )
    ----------------------------
     ( x1 - x2 ) * ( x1 - x3 )

      y2 * ( x - x1 ) * ( x - x3 )
    + ----------------------------
       ( x2 - x1 ) * ( x2 - x3 )

      y3 * ( x - x1 ) * ( x - x2 )
    + ----------------------------
       ( x3 - x1 ) * ( x3 - x2 )

    Out>

##### See also:

Subst .

###

Fibonacci -- Fibonacci sequence

Standard math library

##### Calling format:

    Fibonacci(n)

##### Parameters:

n - an integer

##### Description:

This command calculates and returns the "n"-th Fibonacci number.

The Fibonacci sequence is 1, 1, 2, 3, 5, 8, 13, 21, ... where every number is the sum of the two preceding numbers. Formally, it is defined by F(1)=1, F(2)=1, and F(n+1)=F(n)+F(n-1), where F(n) denotes the n-th Fibonacci number.

##### Examples:

    In> Fibonacci(4)
    Out> 3;
    In> Fibonacci(8)
    Out> 21;
    In> Table(Fibonacci(i), i, 1, 10, 1)
    Out> {1,1,2,3,5,8,13,21,34,55};